TITLE: Derivative of $\frac{x}{f(x)}\frac{df}{dx}$
QUESTION [6 upvotes]: Suppose we have a function $f(x):\mathbb R^+\to\mathbb R^+$ that satisfies:
1) $0\leq\frac{df}{dx} \leq 1$
2) $f(0) = 0$,
then do we have 
$$\frac{d}{dx}\left(\frac{x}{f(x)}\frac{df}{dx} \right)\leq 0?$$

REPLY [4 votes]: This doesn't look right near $0$. Take for example $y = \frac{x + x^2}{2}$, $0 \leq x \leq 0.25$ - This satisfies everything near $0$. But $(\frac{xy'}{y})' = \frac{1}{1+x^2} > 0$.